On pages 20-21 of John Kruschke's Doing Bayesian Data Analysis book (2nd ed.), there is an introductory illustration of Bayesian inference. We know that balls can have four sizes: 1, 2, 3 and 4, but the manufacturing process is not perfect, so the empirically observed balls are of sizes 1.77, 2.23 and 2.7.

Now, the assumptions are:

the three observed balls were produced as balls of the same nominal size,

the priors in this example are 0.25 for each ball type,

the distribution of ball size variability is normal and centered on ball size for each ball type.

We would like to measure which nominal size is most probable for this sample: 1, 2, 3 or maybe 4.

At this stage of the book, this problem is only used to show how posterior changes when data is observed, and no calculations are provided. The Author does offer precisely stated result, though:

...there is 56% probability that the balls are size 2, 31% probability
that the balls are size 3, 11% probability that the balls are size 1,
and only 2% probability that the balls are size 4.

I tried to replicate this result and I failed. My logic was as follows:

$\begingroup$I think, you are doing everything correctly. Not knowing the variance seems to be the problem. Assuming $1$ is quite big from a modelling point of view; $1$ is also the distance between two nominal sizes.$\endgroup$
– JonasJan 12 at 22:39

2

$\begingroup$Found by trial-and-error: if you change scale=1.16, the numbers agree. Direct link to the example: books.google.pl/…$\endgroup$
– Tim♦Jan 12 at 22:56